On the number of Sudoku squares

We provide an upper bound on the number of $n^2\times n^2$ Sudoku squares, and explain intuitively why there is reason to believe that the bound is tight up to a multiplicative factor of a much smaller order of magnitude. A similar bound is established for Sudoku squares with rectangular regions.     

Recursive construction of non-cyclic pandiagonal Latin squares

The pandiagonal Latin squares constructed using Hedayat’s method are cyclic. During the last decades several authors have described methods for constructing pandiagonal Latin squares which are semi-cyclic. In this paper we propose a recursive method for constructing non-cyclic pandiagonal Latin squares of any given order n$n$, where n$n$ is a positive composite integer not divisible by 2 or 3. We also investigate the orthogonality properties of the constructed squares and extend our method to construct non-cyclic pandiagonal Sudoku.     

Construction of Sudoku designs and Sudoku-based uniform designs

In this paper, an easy and effective construction method of Sudoku designs with any order is provided based on the right shift operator. Based on the constructed Sudoku designs, a class of Sudoku-based uniform designs is constructed. Moreover, the properties of the constructed Sudoku designs and Sudoku-based uniform designs are investigated, it is shown that both the constructed Sudoku designs and Sudoku-based uniform designs are uniform designs in terms of discrete discrepancy.     

An efficient algorithm for enumeration of triangulations

We consider the problem of enumerating triangulations of n points in the plane in general position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in O(loglogn) time per triangulation. It improves the previous bound by almost linear factor.     

Maximization of submodular functions: Theory and enumeration algorithms

Submodular functions are powerful tools to model and solve either to optimality or approximately many operational research problems including problems defined on graphs. After reviewing some long-standing theoretical results about the structure of local and global maxima of submodular functions, Cherenin’s selection rules and his Dichotomy Algorithm, we revise the above mentioned theory and show that our revision is useful for creating new non-binary branching algorithms and finding either approximation solutions with guaranteed accuracy or exact ones.     

Orthogonal factor‐pair Latin squares of prime‐power order

We introduce a linear method for constructing factor‐pair Latin squares of prime‐power order and we identify criteria for determining whether two factor‐pair Latin squares constructed using this linear method are orthogonal. Then we show that families of pairwise mutually orthogonal diagonal factor‐pair Latin squares exist in all prime‐power orders.     

Counting and enumeration complexity with application to multicriteria scheduling

In this paper we tackle an important point of combinatorial optimisation: that of complexity theory when dealing with the counting or enumeration of optimal solutions. Complexity theory has been initially designed for decision problems and evolved over the years, for instance, to tackle particular features in optimisation problems. It has also evolved, more or less recently, towards the complexity of counting and enumeration problems and several complexity classes, which we review in this paper, have emerged in the literature. This kind of problems makes sense, notably, in the case of multicriteria optimisation where the aim is often to enumerate the set of the so-called Pareto optima. In the second part of this paper we review the complexity of multicriteria scheduling problems in the light of the previous complexity results. This paper appeared in 4OR 3(1), 1–21, 2005.     

An algorithm for the enumeration of spanning trees

Enumeration of spanning trees of an undirected graph is one of the graph problems that has received much attention in the literature. In this paper a new enumeration algorithm based on the idea of contractions of the graph is presented. The worst-case time complexity of the algorithm isO(n+m+nt) wheren is the number of vertices,m the number of edges, andt the number of spanning trees in the graph. The worst-case space complexity of the algorithm isO(n ²). Computational analysis indicates that the algorithm requires less computation time than any other of the previously best-known algorithms.     

A regional Kronecker product and multiple-pair latin squares

We introduce a variant of the Kronecker product, called the regional Kronecker product, that can be used to build new, larger multiple-pair latin squares from existing multiple-pair latin squares. We present applications to the existence and orthogonality of multiple-pair latin squares.     

Divisors of the number of Latin rectangles

A k×n Latin rectangle on the symbols {1,2,…,n} is called reduced if the first row is (1,2,…,n) and the first column is T(1,2,…,k). Let Rk,n be the number of reduced k×n Latin rectangles and m=⌊n/2⌋. We prove several results giving divisors of Rk,n. For example, (k−1)! divides Rk,n when k?m and m! divides Rk,n when m<k?n. We establish a recurrence which determines the congruence class of for a range of different t. We use this to show that Rk,n≡((−1)^k−1(k−1)!)ⁿ⁻¹. In particular, this means that if n is prime, then Rk,n≡1 for 1?k?n and if n is composite then if and only if k is larger than the greatest prime divisor of n.     

An enumeration of equilateral triangle dissections

We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. We computationally prove Tutte’s conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20.     

Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface Sg of genus g grows asymptotically like

c(g)n5(g−1)/2−1γnn!     

Direct Constructions for General Families of Cyclic Mutually Nearly Orthogonal Latin Squares

Two Latin squares and , of even order n with entries , are said to be nearly orthogonal if the superimposition of L on M yields an array in which each ordered pair , and , occurs at least once and the ordered pair occurs exactly twice. In this paper, we present direct constructions for the existence of general families of three cyclic mutually orthogonal Latin squares of orders , , and . The techniques employed are based on the principle of Methods of Differences and so we also establish infinite classes of “quasi‐difference” sets for these orders.     

GROUP THEORY METHODFOR ENUMERATION OFOUTERPLANAR GRAPHS

1.IntroductionAplanargraphiscalledanouterplanargraph[']ifinitsplaneembeddingitsvenicescanbeplacedontheboundaryofaface.Thisfaceisusuallycalledanouterface.Anouterplanargraphissaidtobemaximumifwecannotaddanyedgetokeepitsouterplanarity.Wesupposethatallouterplanargraphsinvestigatedinthispaperaretwoconnected.Theedgesontheboundaryoftheouterfacearecalledouteredgesandotheredgesarecalledinneredgesorchords.Apathu--vconsistedofouteredgeswithd(u)23andd(v)23iscalledasinglechain.Asinglechainissaidtobetrivi…     

Classification and enumeration of matched free response matrices of linear finite automata

IfH andG are the transfer function matrix and the free response matrix of a linear finite automaton, thenG is called a free response matrix matched toH. GivenH, letG(H) be the set of all possible free response matrices matched toH. A classification and an enumeration on the setG(H) are given in this paper. Supported by National Natural Science Foundation     

Completions of ‐Dense Partial Latin Squares

A classical question in combinatorics is the following: given a partial Latin square P, when can we complete P to a Latin square L? In this paper, we investigate the class of ε‐dense partial Latin squares: partial Latin squares in which each symbol, row, and column contains no more than ‐many nonblank cells. Based on a conjecture of Nash‐Williams, Daykin and Häggkvist conjectured that all ‐dense partial Latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a novel technique derived from a paper by Jacobson and Matthews on generating random Latin squares, and use this technique to study ε‐dense partial Latin squares that contain no more than filled cells in total. In this paper, we construct completions for all ε‐dense partial Latin squares containing no more than filled cells in total, given that . In particular, we show that all ‐dense partial Latin squares are completable. These results improve prior work by Gustavsson, which required , as well as Chetwynd and Häggkvist, which required , n even and greater than 10⁷.